Finite Group Databasebeta

← #36382960

Subgroups — #36382960

Order 22599.

980 conjugacy classes of subgroups; only the 268 normal subgroups are shown.

normal subgroups only

This group has 980 conjugacy classes of subgroups — too many to draw. Only its 268 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21419435C313yesmaximal, complemented
2C32×#11719687C311yesmaximal, characteristic, complemented
3#21419421C314yesmaximal, complemented
4#21419417C316yesmaximal, complemented
5#21419416C312yesmaximal, complemented
6#21419414C316yesmaximal, complemented
7#21419408C312yesmaximal, complemented
8#21419404C311yesmaximal, characteristic, complemented
9#21419398C313yesmaximal, complemented
10#21419391C312yesmaximal, complemented
11#21419387C316yesmaximal, complemented
12AΣL(1,𝔽27)×C31⋊C3C313yesmaximal, complemented
13C31×#6908C311yesmaximal, characteristic, complemented
14C33×C31⋊C3C3211yescharacteristic
15#14578913—312no—
16#14578913C32112yescomplemented
17#14578912—36no—
18#14578912C3216yescomplemented
19#14578911C3216yescomplemented
20C3×#11719687—36no—
21C3×#11719687—33no—
22C3×#11719687—33no—
23C3×#11719687C3211yescharacteristic
24C3×#11719687C3211yescharacteristic
25C3×#11719687C3211yescharacteristic
26C3×#11719687C3212yescomplemented
27C3×#11719687C3212yescomplemented
28C3×#11719687C3212yescomplemented
293−1+2×C31⋊C3—36no—
303−1+2×C31⋊C3C3216yes—
31H(3,𝔽3)×C31⋊C3C3213yes—
32C32×C31⋊C9C3212yescomplemented
33#14578902C3214yescomplemented
34#14578901C32112yescomplemented
35#14578900C3212yescomplemented
36#14578899C3216yescomplemented
37#14578898C32112yescomplemented
38#14578897C3216yescomplemented
39#14578896C3216yescomplemented
40#14578895C32112yescomplemented
41#14578894C3216yescomplemented
42#14578893C3216yescomplemented
43C31×#737C3213yescomplemented
44C32×C9×C31C3211yescharacteristic, complemented
45C31×#733C3212yescomplemented
46C31×#732C3211yescharacteristic, complemented
47C31×#731C3213yescomplemented
48C31×AΣL(1,𝔽27)C3213yescomplemented
49C33×C31C3311yescharacteristic
50C32×C31⋊C3—33no—
51C32×C31⋊C3—33no—
52C32×C31⋊C3—93no—
53C32×C31⋊C3C3312yes—
54C32×C31⋊C3C3311yescharacteristic
553−1+2×C31—36no—
563−1+2×C31C3316yes—
57H(3,𝔽3)×C31C3313yes—
58C3×C9×C31—32no—
59C3×C9×C31—31no—
60C3×C9×C31—33no—
61C3×C9×C31C3311yescharacteristic
62C3×C9×C31C3311yescharacteristic
63C3×C9×C31C3311yescharacteristic
64C32⋊(C31⋊C3)—36no—
65C32⋊(C31⋊C3)C3316yes—
66#11719689—312no—
67#11719689C33112yes—
68C3×C31⋊C9—36no—
69C3×C31⋊C9—36no—
70C3×C31⋊C9C3312yes—
71C3×C31⋊C9C3312yes—
72C3×C31⋊C9C3312yes—
73#11719687—36no—
74#11719687—33no—
75#11719687—33no—
76#11719687—34no—
77#11719687—32no—
78#11719687—36no—
79#11719687—36no—
80#11719687—36no—
81#11719687—312no—
82#11719687H(3,𝔽3)16yes—
83#11719687H(3,𝔽3)13yes—
84#11719686—36no—
85#11719686—36no—
86#11719686—312no—
87C9×C31⋊C3—36no—
88C9×C31⋊C3—36no—
89C9×C31⋊C3—312no—
90#10624934—311nomaximal, Sylow
91C32×C31—33no—
92C32×C31—31no—
93C32×C31—33no—
94C32×C31C3411yescharacteristic
95C3×C31⋊C3—36no—
96C3×C31⋊C3—32no—
97C3×C31⋊C3—91no—
98C3×C31⋊C3—99no—
99C3×C31⋊C3—92no—
100C3×C31⋊C3—273no—
101C3×C31⋊C3C3×H(3,𝔽3)13yes—
102C9×C31—36no—
103C9×C31—31no—
104C9×C31—31no—
105C9×C31C3×H(3,𝔽3)12yes—
106C9×C31C3×H(3,𝔽3)11yescharacteristic
107C31⋊C9—312no—
108C31⋊C9—32no—
109C31⋊C9—32no—
110C31⋊C9C3×H(3,𝔽3)16yes—
1113−1+4—313no—
112C32×3−1+2—311no—
113#6913—316no—
114#6913—314no—
115#6913—312no—
116#6913—316no—
117#6912—312no—
118#6911—311no—
119#6910—313no—
120#6908—312no—
121#6908—316no—
122#6908C31⋊C311yescharacteristic, complemented
123C3×AΣL(1,𝔽27)—313no—
124C3×C31—33no—
125C3×C31—31no—
126C3×C31—93no—
127C3×C31C32×H(3,𝔽3)11yescharacteristic
128C31⋊C3—33no—
129C31⋊C3—92no—
130C31⋊C3—96no—
131C34—311no—
132#737—3112no—
133#737—316no—
134#737—316no—
135#737—936no—
136#737—9312no—
137#737C3×C31⋊C313yescomplemented
138C3×3−1+2—311no—
139C3×3−1+2—312no—
140C3×3−1+2—312no—
141C3×3−1+2—311no—
142C3×3−1+2—311no—
143C3×3−1+2—312no—
144C3×3−1+2—316no—
145C3×3−1+2—933no—
146C3×3−1+2—936no—
147C3×3−1+2—933no—
148C3×3−1+2—936no—
149C3×H(3,𝔽3)—313no—
150C32×C9—312no—
151C32×C9C3×C31⋊C311yescharacteristic, complemented
152#733—3112no—
153#733—314no—
154#733C3×C31⋊C312yescomplemented
155#732—316no—
156#732—312no—
157#732C3×C31⋊C311yescharacteristic, complemented
158#731—3112no—
159#731—316no—
160#731—316no—
161#731C3×C31⋊C313yescomplemented
162AΣL(1,𝔽27)—316no—
163AΣL(1,𝔽27)—3112no—
164AΣL(1,𝔽27)—316no—
165AΣL(1,𝔽27)C3×C31⋊C313yescomplemented
166C31#1062493411yescharacteristic, complemented, Sylow
167C33—2793no—
168C33—311no—
169C33—312no—
170C33—933no—
171C33—933no—
172C33C32×C31⋊C311yescharacteristic
1733−1+2—36no—
1743−1+2—313no—
1753−1+2—316no—
1763−1+2—3112no—
1773−1+2—9312no—
1783−1+2—936no—
1793−1+2—936no—
1803−1+2—9312no—
1813−1+2—936no—
1823−1+2—933no—
1833−1+2—933no—
1843−1+2—936no—
1853−1+2—9312no—
1863−1+2—936no—
1873−1+2—932no—
1883−1+2—934no—
1893−1+2—936no—
1903−1+2C32×C31⋊C316yes—
191H(3,𝔽3)—316no—
192H(3,𝔽3)—936no—
193H(3,𝔽3)C32×C31⋊C313yes—
194C3×C9—31no—
195C3×C9—32no—
196C3×C9—33no—
197C3×C9—312no—
198C3×C9—312no—
199C3×C9—312no—
200C3×C9—9312no—
201C3×C9—936no—
202C3×C9—936no—
203C3×C9—936no—
204C3×C9—936no—
205C3×C9C32×C31⋊C311yescharacteristic
206C3×C9C32×C31⋊C311yescharacteristic
207C3×C9C32×C31⋊C311yescharacteristic
208C32—33no—
209C32—31no—
210C32—33no—
211C32—2792no—
212C32—2799no—
213C32—2791no—
214C32—313no—
215C32—8373no—
216C32—936no—
217C32—932no—
218C32C33×C31⋊C311yescharacteristic
219C9—31no—
220C9—31no—
221C9—36no—
222C9—316no—
223C9—932no—
224C9—932no—
225C9—9312no—
226C9H(3,𝔽3)×C31⋊C311yescharacteristic
227C9H(3,𝔽3)×C31⋊C312yes—
228C3—31no—
229C3—33no—
230C3—93no—
231C3—2792no—
232C3—2796no—
233C3—933no—
234C3C3×H(3,𝔽3)×C31⋊C311yescharacteristic
subgroupG/H
centerC3C3×H(3,𝔽3)×C31⋊C3
commutator, cosocleC32×C31C34
FrattiniC32C33×C31⋊C3
FittingC31×#6908C3
socleC3×C31C32×H(3,𝔽3)
hypercenter#6908C31⋊C3
hypocenterC31#10624934
LCSG ▷ C32×C31 ▷ C3×C31 ▷ C31
UCSC1 ◁ C3 ◁ C3×C9 ◁ #6908

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #21419435C3semidirect
2. C32×#11719687C3semidirect
3. #21419421C3semidirect
4. #21419417C3semidirect
5. #21419416C3semidirect
6. #21419414C3semidirect
7. #21419408C3semidirect
8. #21419404C3semidirect
9. #21419398C3semidirect
10. #21419391C3semidirect
11. #21419387C3semidirect
12. AΣL(1,𝔽27)×C31⋊C3C3semidirect
13. C31×#6908C3semidirect
16. #14578913C32semidirect
18. #14578912C32semidirect
19. #14578911C32semidirect
26. C3×#11719687C32semidirect
27. C3×#11719687C32semidirect
28. C3×#11719687C32semidirect
32. C32×C31⋊C9C32semidirect
33. #14578902C32semidirect
34. #14578901C32semidirect
35. #14578900C32semidirect
36. #14578899C32semidirect
37. #14578898C32semidirect
38. #14578897C32semidirect
39. #14578896C32semidirect
40. #14578895C32semidirect
41. #14578894C32semidirect
42. #14578893C32semidirect
43. C31×#737C32semidirect
44. C32×C9×C31C32semidirect
45. C31×#733C32semidirect
46. C31×#732C32semidirect
47. C31×#731C32semidirect
48. C31×AΣL(1,𝔽27)C32semidirect
122. #6908C31⋊C3semidirect
137. #737C3×C31⋊C3semidirect
151. C32×C9C3×C31⋊C3semidirect
154. #733C3×C31⋊C3semidirect
157. #732C3×C31⋊C3semidirect
161. #731C3×C31⋊C3semidirect
165. AΣL(1,𝔽27)C3×C31⋊C3semidirect
166. C31#10624934semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214194353
C32×#117196873
#214194213
#214194173
#214194163
#214194143
#214194083
#214194043
#214193983
#214193913
#214193873
AΣL(1,𝔽27)×C31⋊C33
C31×#69083
#1062493431